*J. Japan Statist. Soc.,* Vol. 36 (No. 1), pp. 37-62, 2006

## Density Estimation of Lévy Measures for Discretely Observed Diffusion Processes with Jumps

Yasutaka Shimizu

**Abstract. **
We study a nonparametric estimation of Lévy measures for multidimensional jump-diffusion models from some discrete observations. We suppose that the jump term is driven by a Lévy process with finite Lévy measure, that is, a compound Poisson process. We construct a kernel-estimator of the Lévy density under a sampling scheme where the terminal time tends to infinity and at the same time the distance between the observations tends to zero fast enough, and show the *L*^{2}-consistency and the optimal rate in the MSE sense. First, we consider the case where the observations are given continuously and then compare it to the discretely observed case.

*Key words and phrases*:
Consistency, discrete observations, jump-diffusion, kernel density estimation, Lévy density, MSE, optimal rate.

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